Tidal dissipation in multi-planet systems and constraints on orbit fitting

We present here in full detail the linear secular theory with tidal damping that was used to constrain the fit to the HD 10180 planetary system in Lovis et al. (2011, A&A, 528, A112). The theory is very general and can provide some intuitive understanding of the final state of a planetary system when one or more planets are close to their central star. We globally recover the results of Mardling (2007, MNRAS, 382, 1768). However, for the HD 209458 planetary system, we show that the consideration of the tides raised by the central star on the planet leads one to believe that the eccentricity of HD 209458b is most probably much smaller than 0.01